A moving-van rental company uses the polynomial 123.5 + 0.75(m – 190) to calculate the rental charges if a customer drives a van more than 190 miles in one day. In the polynomial, m is the total number of miles that the customer drove the van during the day. Use the Distributive Property to write an equivalent expression for the total cost of renting the van and driving it more than 190 miles in one day.

Answer:

It has no special appearance.

Step-by-step explanation:

Any angle of measure 180° or less is supplementary to some angle. A supplementary angle is one that is the difference between 180° and the angle you have. That is, two supplementary angles total 180°.

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Supplementary angles are readily identifiable in a number of geometries. Adjacent angles of a parallelogram are supplementary; linear angles are supplementary. Same-side interior angles where a transversal crosses parallel lines are supplementary.

the answer is 10. just use the distance formula which is the square root of (x2-x1)^2+(y2-y1)^2

Answer: 10 units

Step-by-step explanation:

The distance  from point (a,b) and (c,d) is given by formula below:-

[tex]D=\sqrt{(c-a)^2+(d-b)^2}[/tex]

Therefore, the distance between the points (5, 1) and (-3, -5) will be :

[tex]D=\sqrt{(-3-5)^2+(-5-1)^2}\\\\\Rightarrow\ D=\sqrt{(-8)^2+(-6)^2}\\\\\Rightarrow\ D=\sqrt{64+36}\\\\\Rightarrow\ D\sqrt{100}\\\\\Rightarrow\ D=10\text{ units}[/tex]

Hence, the distance between the points (5, 1) and (-3, -5) = 10 units

Answer:

Betty's boss use the MEAN to describe the average tip.

Betty use the MEDIAN.

Step-by-step explanation:

The MEAN is 50+45+48+125+85= 353

353/5=70 or 70.6

The MEDIAN is the middle which is 50.

45,48,50,85,125

Hope this helps

Final answer:

The mean tip amount Betty made was $70.60, and the median was $50. Betty's boss used the mean to describe her average nightly tips, whereas Betty used the median.

Explanation:

To calculate the mean and median tip amount for Betty's nightly tips, we list her tips from last week: $50, $45, $48, $125, and $85. To find the mean, we add all the tip amounts together and divide by the number of days she worked. Betty worked 5 days, so the mean is calculated as: (50 + 45 + 48 + 125 + 85) / 5 = $70.60. The median is the middle number when the tips are arranged in order, which are $45, $48, $50, $85, and $125. Thus, the median tip amount is $50.

Based on this information, Betty's boss used the mean to describe her average tip, while Betty herself used the median to describe her average tip.

Answer:

Step-by-step explanation:

When there are many instances of the same calculation, it is convenient to let a spreadsheet or graphing calculator do them. The formula can be entered once and used many times. See the attachment for an example.

1. The surface area of a box with dimensions L, W, D can be written as ...

  S = 2(LW +LD +WD) = 2(LW +D(L+W))

Then the surface area of the left (original) box is ...

  S = 2(2·12 + 8(2+12)) = 2(24 +112) = 272 . . . . square inches

The surface area of the right (proposed) box is ...

  S = 2(4·3 +14(4+3)) = 2(12 +98) = 220 . . . . square inches

The volume of the original box, at 2·12·8 = 192 in³ is greater than the volume of the proposed box (3·4·14 = 168 in³), so the customer gets less cereal with the redesigned box. I don't dig it.

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2. The previous question shows the formula and an example of the calculation. The attachment shows the numbers for this question.

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3. The attachment shows the numbers for this question.

I just took the test!

Answer:

16.2 m²

Answer:

Surface area increases by a factor of 4

Step-by-step explanation:

Given the linear ratio = a : b, then

the area ratio = a² : b²

Here the linear ratio = 1 : 2, hence

area ratio = 1² : 2² = 1 : 4 ← increase by factor of 4

I'll match them for you, but to find the inverse of an equation, all you must do is

[tex]f(x)^{-1}  = 5x[/tex]  →  [tex]f(x) = \frac{x}{5}[/tex]

[tex]f(x)^{-1} = \frac{x^{3}}{2}[/tex]  →  [tex]f(x) = \sqrt[3]{2x}[/tex]

[tex]f(x)^{-1} = x + 10[/tex]  →  [tex]f(x) = x - 10[/tex]

[tex]f(x)^{-1} = \frac{3(x+17)}{2}[/tex]  →  [tex]f(x) = \frac{2x}{3} -17[/tex]

Hope I help ! :)

ANSWER

[tex] \boxed {f(x)= \frac{2x}{3} - 17\to \: f ^{ - 1} (x)=\frac{3x + 51}{2}}[/tex]

[tex] \boxed {f(x) = x - 10 \to {f}^{ - 1} (x) = x + 10 }[/tex]

[tex] \boxed {f(x) = \sqrt[3]{2x} \to {f}^{ - 1} (x) = \frac{ {x}^{3} }{2} }[/tex]

[tex] \boxed {f(x) = \frac{x}{5} \to{f}^{ - 1} (x) = 5x}[/tex]

EXPLANATION

1.

Given :

[tex]f(x) = \frac{2x}{3} - 17[/tex]

Let

[tex]y =\frac{2x}{3} - 17[/tex]

Interchange x and y.

[tex]x=\frac{2y}{3} - 17[/tex]

Solve for y.

[tex]x + 17=\frac{2y}{3} [/tex]

[tex]3x + 51=2y[/tex]

[tex]y=\frac{3x + 51}{2} [/tex]

[tex]f ^{ - 1} (x)=\frac{3x + 51}{2} [/tex]

2.

Given: f(x)=x-10

Let y=x-10

Interchange x and y.

x=y-10

Solve for y.

y=x+10

This implies that,

[tex] {f}^{ - 1} (x) = x + 10[/tex]

3.

Given:

[tex]f(x) = \sqrt[3]{2x} [/tex]

Let

[tex]y=\sqrt[3]{2x} [/tex]

Interchange x and y.

[tex]x=\sqrt[3]{2y} [/tex]

solve for y.

[tex] {x}^{3} = 2y[/tex]

[tex]y = \frac{ {x}^{3} }{2} [/tex]

[tex] {f}^{ - 1} (x) = \frac{ {x}^{3} }{2} [/tex]

4.

Given:

[tex]f(x) = \frac{x}{5} [/tex]

Let

[tex]y = \frac{x}{5} [/tex]

Interchange x and y.

[tex]x = \frac{y}{5} [/tex]

Solve for y.

[tex]y = 5x[/tex]

[tex] {f}^{ - 1} (x) = 5x[/tex]

Answer: A'=(1, 3); B'=(-3, 4);C'=(3, 0); D'=(-2, 5)

You can check the PNG attached as well.

Step-by-step explanation:

You need to represent the symmetry of every given points respet to the line

[tex]y = 2[/tex]

In that case, the line beeing paralell to the x- axis, x- value of the symmetry is the same of the given point and y = 2 is the middle between both points.

Point A(1, 1)

[tex]x_{A} = 1\\ x_{A'} = 1 \\\\\frac{y_{A} +y_{A'} }{2} =2\\y_{A'} = 4 - y_{A} = 4 - 1 = 3[/tex]

Point B(-3, 0)

[tex]x_{B} = 1\\ x_{B'} = 1 \\\\\frac{y_{B} +y_{B'} }{2} =2\\y_{B'} = 4 - y_{B} = 4 - 0 = 4[/tex]

Point C(3, 4)

[tex]x_{C} = 1\\ x_{C'} = 1 \\\\\frac{y_{C} +y_{C'} }{2} =2\\y_{C'} = 4 - y_{C} = 4 - 4 = 0[/tex]

Point D(-2, -1)

[tex]x_{D} = 1\\ x_{D'} = 1 \\\\\frac{y_{D} +y_{D'} }{2} =2\\y_{D'} = 4 - y_{D} = 4 - (-1) = 4 + 1 = 5[/tex]

Answer:

[tex]\displaystyle x=\frac{-8\pm\sqrt{(8)^2-4(4)(-221)}}{2(4)}\ \text{; x = 6.5 and x = -8.5}[/tex]

Step-by-step explanation:

Subtract the right side of the given equation to put it into standard form:

  4x² +8x -221 = 0

Then the coefficients used in the quadratic formula are ...

When these are filled into the form ...

[tex]\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

the result is as shown above.

The angle of elevation from Jonathan to Mina is approximately 43.81 degrees.

If you know the lengths of two sides of a right triangle, you can use trigonometric ratios to calculate the measures of one (or both) of the acute angles.

Here,
We can use the tangent function to find the angle of elevation from Jonathan to Mina.

Let θ be the angle of elevation, then we have:

tan(θ) = opposite / adjacent

Where the opposite is the height of the tower (389 feet) and adjacent is the distance between Jonathan and the base of the tower (412 feet).

So, we have:

tan(θ) = 389 / 412

θ ≈ 43.81 degrees

Therefore, the angle of elevation from Jonathan to Mina is approximately 43.81 degrees.

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To find the tension in each cable, we need to consider the forces acting on the weight. Using Newton's second law and trigonometry, we can set up equations to solve for the tension in each cable. The tension in each cable is (OPTION C) 50 lb.

To find the tension in each cable, we need to consider the forces acting on the weight. In this case, there are two cables pulling upward and the weight pulling downward. Since the weight is in equilibrium, the sum of the upward forces must equal the downward force. Let's label the tension in the first cable as T1 and the tension in the second cable as T2.

Using Newton's second law, we can set up the following equation: T1 + T2 = 100 lb. We also know that the angle between each cable and the vertical is the same, so the horizontal components of tension in both cables are equal. Let's call this component T.

Since the weight is in equilibrium, the vertical components of tension in the cables must also balance the weight's weight. The vertical component of tension in each cable can be found using trigonometry. Let's call this component V.

Now, we can set up two equations for the horizontal and vertical components:

T + T = 100 lb (Equation 1)

V + V = weight of the weight = 100 lb (Equation 2)

Since T1 + T2 = 2T, Equation 1 becomes:

2T = 100 lb

T = 50 lb

Substituting T = 50 lb into Equation 2, we can find V:

2V = 100 lb

V = 50 lb

Therefore, the tension in each cable is 50 lb.

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Answer: Third option.

Step-by-step explanation:

You need to find the surface area with the formula for calculate the surface area of a rectangular prism:

[tex]SA=2(wl + lh + hw)[/tex]

Where w is the width, l is the length, and h is the height.

You can observe in the figure that:

[tex]l=5cm\\w=4cm\\h=2cm[/tex]

Then, substituting the values into the formula  [tex]SA=2(wl + lh + hw)[/tex], you get that  minimum amount of cardboard used for the wrapper is:

[tex]SA=2[(4cm)(5cm)+(5cm)(2cm)+(2cm)(4cm)]\\SA=76cm^2[/tex]

Answer:

third option

Step-by-step explanation:

Answer:

  see below

Step-by-step explanation:

Since there are fewer sequences than functions, we'll identify the matchup according to the sequence.

11, 33, 55, 77, ...

The first term is 11. The terms have a common difference of 33 -11 = 22. That is, each term is 22 more than the previous one. The appropriate recursive function is ...

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-18, -108, -648, -3888, ...

The first term is -18. The terms obviously do not have a common difference, but their common ratio is -648/-108 = -108/-18 = 6. That is, each term is 6 times the previous one. Then the appropriate recursive function is ...

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-18, 3, 24, 45, ...

The first term is -18. The terms have a common difference of 3-(-18) = 21. That is, each term is 21 more than the previous one. The appropriate recursive function is ...

Answer:

11, 33, 55, 77, ...=f(n) = f(n - 1) + 22 for n = 2, 3, 4, ...

-18, -108, -648, -3,888, ...=f(n) = 6 · f(n - 1) for n = 2, 3, 4, ...

-18, 3, 24, 45, ...=f(n) = f(n - 1) + 21 for n = 2, 3, 4, ...

Step-by-step explanation:

Answer:

2u + v - 4w = <40 , -4>

6u - 8v = <-78 , 58>

4v - 7w = <101 , -36>

11u + 3w = <-88 , 89>

Step-by-step explanation:

* Lets find the value of each operation to find its resultant vector

# 2u + v - 4w

∵ u = <-5 , 7>

∴ 2u = <-10 , 14>

∵ v = <6 , -2>

∵ w = <-11 , 4>

∴ 4w = <-44 , 16>

∴ 2u + v - 4w = <-10 + 6 - -44 , 14 + -2 - 16>

∴ 2u + v - 4w = <40 , -4>

# 6u - 8v

∵ u = <-5 , 7>

∴ 6u = <-30 , 42>

∵ v = <6 , -2>

∵8v = <48 , -16>

∴ 6u - 8v = <-30 - 48 , 42 - -16>

∴ 6u - 8v = <-78 , 58>

# 4v - 7w

∵ v = <6 , -2>

∴ 4v = <24 , -8>

∵ w = <-11 , 4>

∴ 7w = <-77 , 28>

∴ 4v - 7w = <24 - -77 , -8 - 28>

∴ 4v - 7w = <101 , -36>

# 11u + 3w

∵ u = <-5 , 7>

∴ 11u = <-55 , 77>

∵ w = <-11 , 4>

∴ 3w = <-33 , 12>

∴ 11u + 3w = <-55 + -33 , 77 + 12>

∴ 11u + 3w = <-88 , 89>

Answer:

60

Step-by-step explanation:

Angles AKB and EKD are vertical angles and congruent.

The measure of angle AKB is the sum of the measures of arcs AB and ED divided by 2.

m<AKB = (110 + 130)/2 = 120

Angles AKB and AKE are supplementary angles, so their measures add to 180 deg.

m<AKE + m<AKB = 180

m<AKE + 120 = 180

m<AKE = 60

Answer:

The range of width is 10 ≤ W ≤ 28

Step-by-step explanation:

* lets study the meaning of compound inequality

- If x is greater than a and x is smaller than b, then x is between a and b

∵ x > a and x < b

∴ The compound inequality is ⇒ a < x < b

# Ex: ∵ x > -2 and x < 10

∴ The compound inequality is ⇒ -2 < x < 10

- If x is greater than or equal a and x is smaller than or equal b, then

 x is from a and b

∵ x ≥ a and x ≤ b

∴ The compound inequality is ⇒ a ≤ x ≤ b

# Ex: ∵ x ≥ -2 and x ≤ 10

∴ The compound inequality is ⇒ -2 ≤ x ≤ 10

* Now lets solve the problem

- The garden in the shape of a rectangle with dimensions length (L)

 and width (W)

- The length of the garden is 10 feet

- The perimeter (P) of the garden is at least 40 feet and not more than

  76 feet

∵ L = 10 feet

∵ P = 2L + 2W

- At least means greater than or equal (≥) and not more than means

 smaller than or equal (≤)

∴ P ≥ 40 feet

∴ P ≤ 76 feet

- lets use the rule of the perimeter

∴ 2(10) + 2(W) ≥ 40 ⇒ simplify

∴ 20 + 2W ≥ 40 ⇒ subtract 20 from both sides

∴ 2W ≥ 20 ⇒ divide both sides by 2

∴ W ≥ 10 ⇒ (1)

- Do similar with P ≤ 76

∴ 2(10) + 2(W) ≤ 76 ⇒ simplify

∴ 20 + 2W ≤ 76 ⇒ subtract 20 from both sides

∴ 2W ≤ 56 ⇒ divide both sides by 2

∴ W ≤ 28 ⇒ (2)

- From (1) and (2)

∴ 10 ≤ W ≤ 28 ⇒ compound inequality

* The range of the width is from 10 feet to 28 feet

Answer:

It is c = kt.

Step-by-step explanation:

This is direct variation . As the time increases the number of cranes increases.

The equation is c = kt  where k is the constant  of variation.  In this case it will be a positive value because c is increasing with time.

If they make 20 cranes in 10 minutes then we can find the value of k by plugging in these values;

20 = k * 10

g = 20/10

k = 2.

So they make 2 cranes per minute.

Answer:

1.)[tex]-12+4x=-12+4x[/tex]

2.) [tex]2-2x \neq -8-2x[/tex]

Step-by-step explanation:

The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding all the products.

[tex]a(b+c)=ab+ac[/tex]

For the first equation

[tex]-2(6-2x)=4(-3+x)\\(-2)(6)+(-2)(-2x)=(4)(-3)+(4)(x)\\-12+4x=-12+4x[/tex]

For the second equation

[tex]5-1(2x+3)=-2(4+x)\\5+((-1)(2x)+(-1)(3))= (-2)(4)+(-2)(x)\\5+(-2x-3) = -8+(-2x)\\5-2x-3=-8-2x\\2-2x \neq -8-2x[/tex]

.

The value 1.8 represents the growth factor in the expression, indicating the monthly growth rate of club members after advertising.

The value 1.8 in the given expression that models the number of club members, in hundreds, after advertising for t months represents the growth factor, which is the rate at which the total number of members at the club increases each month. This is not the initial number of members, nor the monthly percentage increase, nor the number of new members joining per month, but rather the multiplier applied to the previous month's total to find the next month's projected total.

Answer:

Step-by-step explanation:

Set up a ratio in the form of pages/minute:

[tex]\frac{pages}{minute} :[/tex]

then fill in next to that how many pages per minute George can read:

[tex]\frac{pages}{minute}:\frac{15}{12}[/tex]

Now we want to figure out how long (time is our unknown, so we call that x) it will take him to read 85 pages.  The 85 goes in a ratio on the same line as the other number of pages:

[tex]\frac{pages}{minute}:\frac{15}{12}=\frac{85}{x}[/tex]

Cross multiply to get 15x = 1020

Now divide both sides by 15 to get x = 68 minutes, which is an hour and 8 minutes.

Answer:

68 minutes

Step-by-step explanation:

Given that George read 15 pages in a book in 12 minutes, and we need to find out how long it will take him to finish the book if it has 85 pages, we must first find how much minutes it takes to read per page because we need to find the amount of time.

Write an equation (In this case p=pages and t= minutes or time):

(Remember! We are trying to find the minutes per page, not how many pages per minute.)

15p=12t

p=12/15t

p=0.8t

We now know he reads 1 page in 0.8 minutes. Given that we need to find the amount of time to read 85 pages, multiply 0.8 by 85 because 0.8 minutes=1 page.

85*0.8=68

Therefore, it takes him 68 minutes to read 85 pages

Answer:

15 seconds

Step-by-step explanation:

If you make a table of values for the dog and the squirrel using d = rt, then the rates are easy:  the dog's rate is 150 and the squirrel's is 100.  The t is what we are looking for, so that's our unknown, and the distance is a bit tricky, but let's look at what we know:  the dog is 200 feet behind the squirrel, so when the dog catches up to the squirrel, he has run some distance d plus the 200 feet to catch up.  Since we don't know what d is, we will just call it d!  Now it seems as though we have 2 unknowns which is a problem.  However, if we solve both equations (the one for the dog and the one for the squirrel) for t, we can set them equal to each other.  Here's the dog's equation:

d = rt

d+200 = 150t

And the squirrel's:

d = 100t

If we solve both for t and set them equal to each other we have:

[tex]\frac{150}{d+200} =\frac{100}{d}[/tex]

Now we can cross multiply to solve for d:

150d = 100d + 20,000 and

50d = 20,000

d = 400

But we're not looking for the distance the squirrel traveled before the dog caught it, we are looking for how long it took.  So sub that d value back into one of the equations we have solved for t and do the math:

[tex]t=\frac{100}{d}=\frac{100}{400} =\frac{1}{4}[/tex]

That's 1/4 of a minute which is 15 seconds.

Answer:

4 mins

Step-by-step explanation:

Let x = the distance the squirrel runs before it's caught,

then the dog runs 200 + x.

distance/rate = time

x/100 = (200+x)/150 =>x/2 = (200+x)/3 => 400 +2x = 3x => x = 400

The squirrel runs 400' in 4 minutes.

Answer:

  A)  (20, 0)

Step-by-step explanation:

Double the second equation and add that to the first:

  (0.2x +0.5y) +2(-0.1x +0.3y) = (4) +2(-2)

  1.1y = 0

   y = 0

Substitute this value into either equation to find x.

  0.2x +0.5·0 = 4

  x = 4/0.2 = 20

The solution is (x, y) = (20, 0).

Final answer:

The surveying method that is most likely to produce a representative sample of the yogurt shop's customers is true random sampling.

Explanation:

The surveying method that is most likely to produce a representative sample of the yogurt shop's customers is the true random sampling method. This method involves randomly selecting participants from the entire population of yogurt shop customers, ensuring that each customer has an equal chance of being selected. This helps to minimize bias and ensure that the sample is representative of the entire customer population.

For example, the manager can generate a list of all the customers who have made purchases at the yogurt shop over a specific period of time, and then use a random number generator or a random selection method to choose a certain number of customers from the list. This will ensure that the selected participants represent the diversity of the yogurt shop's customer base.

The most appropriate surveying method for the yogurt shop manager to obtain a representative sample of customers is systematic sampling.

To obtain a representative sample of the yogurt shop's customers, the most appropriate surveying method would be a systematic sampling approach. This involves selecting every nth customer who visits the shop during different times of the day and different days of the week.

By using a systematic sampling method, the manager can ensure that the sample includes customers from various demographic groups, such as different age ranges, genders, and visiting patterns. This approach reduces the potential for bias that may arise from convenience sampling methods, where only customers who are readily available or willing to participate are surveyed.

Additionally, the systematic sampling method allows the manager to capture the preferences of both regular and occasional customers, as well as those who visit during peak and off-peak hours. This comprehensive representation of the customer base increases the likelihood that the survey results will accurately reflect the preferences of the yogurt shop's overall customer population.

Systematic sampling is more time-consuming and resource-intensive than convenience sampling, but it is a more reliable method for obtaining a representative sample of the yogurt shop's customers, which is crucial for making informed decisions about introducing new flavors that will appeal to a wide range of customers.

The owner wants people to believe that dance classes are popular so that they sign up for classes. Therefore, option C is the correct answer.

Probability can be defined as the ratio of the number of favourable outcomes to the total number of outcomes of an event.

The most likely reason the owner of the dance studio chain releases the report is to demonstrate the success of their business. By highlighting the fact that participation in dance classes has increased by 5% in each of the past three years, the owner is showing the public that the business is doing well and that they are providing a valuable service. This report may also be used to encourage potential customers to join the studio's classes, which would further increase their profits.

Therefore, option C is the correct answer.

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Answer:

7/50

Step-by-step explanation:

Experimental and theoretical probability are much different.

With experimental, just read the experiment: the number two was rolled 7 times.

Put that over 50.

Avoid questions with theoretical probability, that's where math comes in.

Final answer:

The experimental probability of rolling a two on a number cube that was rolled 50 times and showed the number two 7 times is 0.14 or 14%.

Explanation:

To calculate the experimental probability of the number cube showing the number two, we divide the number of times two appears by the total number of rolls. In this case, the number cube was rolled 50 times and the number two appeared 7 times. Therefore, the experimental probability, denoted as P(2), is calculated as:

P(2) = Number of times two appears / Total number of rolls

P(2) = 7 / 50

P(2) = 0.14

So, the experimental probability of rolling a two on this number cube, based on the given data, is 0.14 or 14%.

Answer:

0.91.

Step-by-step explanation:

We use the formula:

P(a U b) = p(a) + p(b) - p (a ∩ b)     where P(a U b) is p(a or b) and p (a ∩ b) is p  (a and b).

So p(a or b) = 0.78 + 0.66 - 0.53

= 0.91.

Answer:

$9.50 - for adults

$7.50 - children

Step-by-step explanation:

All you have to do is multiply $4.75 by 2 since there are two adults and multiply $2.50 by 3 since there  are three children.

The surface area of a sphere is given by

[tex]S = 4\pi r^2[/tex]

We deduce

[tex]r = \sqrt{\dfrac{S}{4\pi}}[/tex]

So, in your case, the radius is

[tex]r = \sqrt{\dfrac{100\pi}{4\pi}}=\sqrt{25}=5[/tex]

The volume of a sphere is given by

[tex]V=\dfrac{4}{3}\pi r^3[/tex]

So, we have

[tex]V = \dfrac{4}{3} \pi 5^3 = \dfrac{500}{3}\pi[/tex]

Answer:

Sometimes you to plot some points to get a good approximation of the location of extreme values.

Approximating some points, especially influential points or outliers, helps to get a better grasp of the location of the extreme values. This is crucial in fields like calculus, statistics and graphing, and it assists in identifying patterns and trends. Additionally, substitution of 'x' values in an equation can help estimate 'y' values.

Sometimes in mathematics, particularly in fields such as calculus, statistics, and graph theory, you need to approximate some points for a good understanding of the location of extreme values. This practice is especially useful when identifying influential points or outliers within a data set, as these points significantly alter the slope or fitness of a regression line. When you find such points, you can exclude them from your calculations to get a more accurate overview of the general pattern or trend.

In the case of graphing, selection of an appropriate scale for both axes is also crucial. The scale should reflect all your data while also making it easy to identify any trends within it. Too large a scale can make data changes hard to see, while a too fine scale necessitates more space for the graph and can crowd information.

Additionally, in calculus and algebra, to estimate the 'y' values for various 'x-values', one can substitute the 'x' values into the equation. This helps to offer a clearer insight into the correlation of variables within a function.

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Let [tex]x[/tex] be the amount (mL) of the pure vinegar the chef will use, and [tex]y[/tex] the amount of dressing. She wants to end up with a 310 mL mixture, so

[tex]x+y=310[/tex]

For each mL used of the dressing, 0.07 mL is vinegar, and the chef wants to end up with a 19% vinegar mixture, so

[tex]x+0.07y=0.19(x+y)=58.9[/tex]

Now

[tex]x+y=310\implies y=310-x[/tex]

[tex]\implies x+0.07(310-x)=58.9[/tex]

[tex]\implies0.93x+21.7=58.9[/tex]

[tex]\implies0.93x=37.2[/tex]

[tex]\implies x=40[/tex]

[tex]\implies y=270[/tex]